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DISTANCES SETS THAT ARE A SHIFT OF THE INTEGERS AND FOURIER BASIS FOR PLANAR CONVEX SETS (2007)
BibTeX
@MISC{Iosevich07distancessets,
author = {Alex Iosevich and Philippe Jaming},
title = {DISTANCES SETS THAT ARE A SHIFT OF THE INTEGERS AND FOURIER BASIS FOR PLANAR CONVEX SETS},
year = {2007}
}
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Abstract
The aim of this paper is to prove that if a planar set A has a difference set ∆(A) satisfying ∆(A) ⊂ Z + + s for suitable s than A has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth boundary and everywhere non-vanishing curvature, then #(A ∩ [−q, q] 2) ≤ C(K)q where C(K) is a constant depending only on K. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from [IKP01] and [IKT01] that if K is a centrally symmetric convex body with a smooth boundary and
